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A Conjecture on Exceptional Orthogonal Polynomials

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Abstract

Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

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Notes

  1. A wider class of these transformations has been extensively used in quantum mechanics to generate new exactly solvable problems from known ones. The subclass of interest to us in the context of OPS consists of the set of transformations that preserve the polynomial character of the eigenfunctions. This particular class of Darboux transformations was characterized in [12, 13].

  2. We stress that invariance of the whole flag \(\mathcal {U}:U_{1}\subset U_{2}\subset\cdots\) is a much stronger condition than the invariance of the total space \(\mathcal {U}\). For the purpose of this study, we will always require invariance of the flag, since this condition leads to polynomial eigenfunctions of the operator.

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Acknowledgements

The research of DGU was supported in part by MICINN-FEDER grant MTM2009-06973 and CUR-DIUE grant 2009SGR859. The research of NK was supported in part by NSERC grant RGPIN 105490-2011. The research of RM was supported in part by NSERC grant RGPIN-228057-2009.

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Authors and Affiliations

  1. Departamento de Física Teórica II, Universidad Complutense de Madrid, 28040, Madrid, Spain

    David Gómez-Ullate

  2. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 0B2, Canada

    Niky Kamran

  3. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada

    Robert Milson

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  1. David Gómez-Ullate
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  2. Niky Kamran
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  3. Robert Milson
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Correspondence to Niky Kamran.

Additional information

Communicated by Elizabeth Mansfield.

To Peter Olver, with all our friendship and admiration.

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Gómez-Ullate, D., Kamran, N. & Milson, R. A Conjecture on Exceptional Orthogonal Polynomials. Found Comput Math 13, 615–666 (2013). https://doi.org/10.1007/s10208-012-9128-6

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  • DOI: https://doi.org/10.1007/s10208-012-9128-6

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